Re: Theories M,
- From: zuhair <zaljohar@xxxxxxxxx>
- Date: Mon, 15 Jun 2009 19:23:28 -0700 (PDT)
On Jun 15, 5:26 pm, Transfer Principle <lwal...@xxxxxxxxx> wrote:
On Jun 13, 9:58 pm, zuhair <zaljo...@xxxxxxxxx> wrote:
Hi all,
Theories M are a constilation of set theories, in which we have
infinitely many universes one inside the other Uo in U1 in U2
in........ etc. all universes are transitive, every universe is the
set of all hereditary U cardinality sets. By varying the total number
of universes, and the size of each universe we can reach to different
set theories, some are equivalent to ZC, some are to ZFC, some are to
ZFC+infinitly many accessible cardinals, some might indeed contain
Large cardinals even higher than inaccessible ones. So they can surve
as a good theories for axiomatizing Large cardinals.
OK, sounds interesting so far.
P is a "sequence defining predicate on universals" only if it
fulfills the following conditions:
1) Px -> x is universal
2) Pxy -> (x in y or y in x)
Notice that zuhair first uses P as a one-place predicate symbol to
write "Px", then as a two-place predicate to write "Pxy." I assume
what zuhair really means is:
2) (Px & Py) -> (xey v yex)
3) Exist x (Px and not Exist y (Py and y in x))
4) Px -> Exist y (y is P_successor_of x)
w[h]ere:
y is P_successor_of x iff
[Py and x in y and not exist z (Pz and x in z and z in y)]
if P fulfills(in addition to be the above four conditions) a fifth
condition of allowing a predecessor for each P(x) as defined below,
then P would be called a "perfect sequence defining predicate on
universals"
the fifth condition is:
5) (Px & Exist z( Pz and z in x)) ->
Exist y (y is P_Predecessor_of x)
w[h]ere:
y is P_Predecessor_of x iff
[Py and y in x and not exist z (Pz and y in z and z in x)]
We know that zuhair's intention is for the relation "e" (or "in"), as
restricted to the objects satisfying P, to be a wellorder. We know
this because he intends U0 to be the set of all hereditarily finite
sets, then U1 to be the set of all hereditarily countable sets, and
so on. In an earlier thread, Aatu asserted zuhair that all of these
sets exist in ZFC. There is actually a notation for this: H_kappa
is the set of all sets hereditarily less than cardinality kappa. So
we can write zuhair's intention as:
U_alpha = H_aleph_alpha
for each ordinal alpha. Then since the ordinals are wellordered, we
would want "e" restricted to sets satisfying P to be a wellorder.
But do 1) - 5) guarantee this? Let's see. First, we need "e" to be a
(strict) partial order. It appears that axiom 5), namely:
5)Size limitation: Every uniserval only contains subsets of it that
are subnumerous to it.
forall x ( x is universal ->
forall y ( y in x iff (y subset of x and y subnumerous to x)))
"subnumerous to" is defined in the conventional manner.
is required to show this. Axiom 5) guarantees that "e" is identical
to this "subnumerosity" relation, which I'll write as "<", on the sets
satisfying P. Then since "<" is a strict partial (indeed, a strict
total)
order on these sets, so is "e".
Regarding zuhair's relation "P_successor_of" -- let x be the set of
hereditarily countable sets. Now x is an element of another universal
set, y, but what set is it? To determine this, we need to know what
card(x) is, but Aatu, who told us earlier that the set x exists,
didn't
tell us what card(x) is.
Let's see. We see that every countable ordinal is evidently a
hereditarily countable set. Therefore, there exist at least as many
hereditarily countable sets as there are countable ordinals, and we
know that there are exactly aleph_1 countable ordinals. So we know
that card(x) is at least aleph_1, but that doesn't tell us what card
(x)
exactly is. I wonder the exact value of card(x) depends on such
axioms as AC or CH.
Notice that if the set of hereditarily countable sets has cardinality
aleph_2 (or greater), then it is not an element of the set of all sets
with cardinality at most aleph_1. Then the latter can't be the
P_successor_of the former, since the former isn't even an element
of the latter.
In order for zuhair's universes to correspond to the H_kappa's (i.e.,
for U_alpha to equal H_aleph_alpha) as zuhair intends, it's
necessary for card(H_kappa) to be exactly kappa. It's easy to show
that card(H_kappa) >= kappa. Perhaps Aatu, who seems to be an
expert on these H_kappa's in the first place, could please enlighten
us on what card(H_kappa) is, or whether its value depends on such
axioms as AC or GCH.
Even if we knew that card(H_kappa) = kappa, rules 4) and 5) still
don't give us a wellorder. Indeed, rule 5) requires that universes
must
have _predecessors_ as well as successors, even though ordinals
don't always have predecessors.
Let U_0 be the smallest universe, which exists by rule 3). By rule
4), we define successor universes U_1, U_2, U_3, U_4, U_5, and
so on. By axiom 3), which is:
3)schema of Limit Universal : if P is a prefect sequence defining
predicate on univesals then, all closures of:
Px -> Exist y ( y is a universal and x in y and
forall z (z is P_successor_of x -> z in y))
are axioms.
the perfect sequence U_0, U_1, U_2, etc., must have a limit
universe, which we can call U_omega. Then by rule 4), we can
define successor universes U_(omega+1), U_(omega+2), etc.
But notice that U_omega _can't_ have a predecessor universe. For
suppose its predecessor U_(omega-1) existed. By the definition of
predecessor, U_(omega-1) is an element of U_omega. Since by
the definition of U_omega as a _limit_ in schema 3), U_omega is
closed under successor, so if U_(omega-1) is an element of
U_omega, then so is its successor, U_omega itself. But no universe
U_omega can contain itself as an element by axiom 5) -- since
U_omega can't be subnumerous to _itself_. Therefore, we conclude
that U_omega has no successor.
Can we prove that U_(omega+omega) exists? No -- it's because
schema 3) only guarantees that _perfect_ sequences of universes
have limits, and by definition, every universe in a perfect sequence
except U_0 must have a predecessor, yet U_omega obviously has
no predecessor. Thus schema 3) can't be used to prove that the
universe U_(omega+omega) exists (much less U_theta for some
large cardinal theta).
Strong point. Actually I was thinking of that, and that is true, my
theories that were presented in this thread do not prove the existence
of U_theta were theta is a large cardinal, I didn't come yet to this
point, However, it can be inflated to that degree, I have an axiom in
my mind, but I am not sure if it is acceptable in set theories, and I
call it the
Axiom of Inflation: For every set x there exist a set of unviersals
that is supernumerous to it.
In symboles:
Forevery set x Exist a set y ( forall z (z in y ->z is a universal )
and y supernumerous to x ).
This will guaranty U_theta where theta is some Large cardinal.
However this is another subject.
Of course, there's no reason that the universes U_alpha must
look like H_aleph_alpha. Indeed, some so-called "cranks" have
expressed the desire for infinity to have "predecessors." Also,
some "cranks" such as tommy1729 have expressed the desire to
have an upper limit to set size, so that a universe such as zuhair's
U_omega (or U_aleph_0) is already the set V of all sets. But the
point is that I don't think this is what _zuhair_ desires.
.
- References:
- Theories M,
- From: zuhair
- Re: Theories M,
- From: Transfer Principle
- Theories M,
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